Past exam of the physics B course of the University of Cambridge / 2014 / Paper 2 / Section C / C10 / a

Suppose an engine were more efficient than a reversible Carnot engine between the same reservoirs. Couple it to a reversed Carnot engine, scaling the cycles to have equal work. With no net external work, their combined effect would be to transfer heat from the colder reservoir to the hotter one, contradicting the Second law of thermodynamics. No such engine can exceed ${\eta }_C=1-T_c/T_h$. Equivalently, the cyclic entropy inequality gives $-Q_h/T_h+Q_c/T_c\ge 0$, hence the same bound.